Average Error: 43.8 → 0.8
Time: 30.1s
Precision: 64
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))
↓
\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))
double f(double x, double y) {
        double r49864 = x;
        double r49865 = exp(r49864);
        double r49866 = -r49864;
        double r49867 = exp(r49866);
        double r49868 = r49865 + r49867;
        double r49869 = 2.0;
        double r49870 = r49868 / r49869;
        double r49871 = y;
        double r49872 = cos(r49871);
        double r49873 = r49870 * r49872;
        double r49874 = r49865 - r49867;
        double r49875 = r49874 / r49869;
        double r49876 = sin(r49871);
        double r49877 = r49875 * r49876;
        double r49878 = /* ERROR: no complex support in C */;
        double r49879 = /* ERROR: no complex support in C */;
        return r49879;
}
↓
double f(double x, double y) {
        double r49880 = x;
        double r49881 = exp(r49880);
        double r49882 = -r49880;
        double r49883 = exp(r49882);
        double r49884 = r49881 + r49883;
        double r49885 = 2.0;
        double r49886 = r49884 / r49885;
        double r49887 = y;
        double r49888 = cos(r49887);
        double r49889 = r49886 * r49888;
        double r49890 = 0.3333333333333333;
        double r49891 = 3.0;
        double r49892 = pow(r49880, r49891);
        double r49893 = 0.016666666666666666;
        double r49894 = 5.0;
        double r49895 = pow(r49880, r49894);
        double r49896 = 2.0;
        double r49897 = r49896 * r49880;
        double r49898 = fma(r49893, r49895, r49897);
        double r49899 = fma(r49890, r49892, r49898);
        double r49900 = r49899 / r49885;
        double r49901 = sin(r49887);
        double r49902 = r49900 * r49901;
        double r49903 = /* ERROR: no complex support in C */;
        double r49904 = /* ERROR: no complex support in C */;
        return r49904;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 43.8

    \[\Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Taylor expanded around 0 0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\frac{1}{3} \cdot {x}^{3} + \left(\frac{1}{60} \cdot {x}^{5} + 2 \cdot x\right)}}{2} \cdot \sin y i\right))\]

  3. Simplified0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}}{2} \cdot \sin y i\right))\]

  4. Final simplification0.8

    \[\leadsto \Im(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{\mathsf{fma}\left(\frac{1}{3}, {x}^{3}, \mathsf{fma}\left(\frac{1}{60}, {x}^{5}, 2 \cdot x\right)\right)}{2} \cdot \sin y i\right))\]

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y)
  :name "Euler formula imaginary part (p55)"
  :precision binary64
  (im (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))